Mathematical Computation December 2013, Volume 2, Issue 4, PP.73-80

Numerical Simulation of 3D Thermal-Fluid Coupled Model in Porous Medium Xiangrui Chen 1,2, Tangwei Liu 1,2,# 1.Fundamental Science on Radioactive Geology and Exploration Technology Laboratory, East China Institute of Technology, NanChang, Jiangxi, 330013, China 2.School of Science, East China Institute of Technology, NanChang, Jiangxi, 330013, China # Email: twliu@ecit.cn

Abstract This article has presented finite difference schemes of 3D coupled model of seepage field and temperature field based on Darcyâ€™s seepage law and heat conduction rule in porous media, and gave the corresponding algorithm and main codes of the numerical computation. Numerical results showed that the numerical method of 3D coupled model is very efficient. Keywords: 3D Thermal-Fluid Coupled Model; Difference Scheme; Numerical Simulation; Porous Medium

1 INTRODUCTION The basic law of flow in porous medium was firstly presented by Mr. Darcy who was a famous French hydraulics in the middle of the 19th century. At present, the numerical methods of the hydrogeology and heat transfer models in porous media and rock matrix-fractured media are popular concentrated [1, 2, 3, 4, 7, 9]. The numerical simulations methods for these 1D and 2D models have made great progress, respectively [4, 8, 9]. And numerical methods for coupled model such as Burgersâ€™ equations have been presented in [10, 11, 12]. However, the numerical simulation practice of 3D thermal-fluid coupled model in porous medium still is difficult to be solved. In this paper, finite difference scheme of 3D thermal-fluid coupled model and compare the numerical results have been presented with the exact solution by the numerical example.

2 THREE DIMENSIONAL THERMAL-FLUID COUPLED MODEL Considering the problem that fluids flow in porous media, the factors that cause the change of the water head include heat convection, heat transfer effect, mechanical dispersion, heat exchange and so on[ 2]. By the principle of conservation of energy, we have (1) The symbols n, , Ts and Tw denote the porosity of media, specific heat of water, the solidus temperature and the water temperature, respectively. The symbol denotes hydrodynamic dispersion coefficient, and denotes heat transfer coefficient between solid phase and the aqueous phase. And based on the Darcy's law [1], we can get , Where permeability coefficient of porous media,

(2)

represents the seepage velocity, K is the represents the water head value.

Seepage continuity equation can be expressed as . - 73 www.ivypub.org/MC

(3)

Since the analytical solution of equations (1)-(3) is usually difficult to seek out, its approximate solution is found out by finite difference method.

3 DIFFERENCE SCHEMES OF THE HEAT TRANSFER AND CONTINUITY EQUATIONS Since equation (1) is very difficult to calculate, some assumptions are made to simple the equation. It is assumed that the porosity n is a const, the specific heat of water is a constant, and the solidus temperature Ts is equal to the water temperature Tw. Then the above equation (1) can be simplified as the following equation .

(4)

Omitting all the subscripts, the equation (4) is expressed as .

(5)

Difference scheme of equation (5) is + .(6) In the formula (6), set

.Then the formula (6) can be simplified to

(7)

Using the centered-difference formula, the equation (3) at the points

can be expressed as

+ Set

(8)

, the formula (8) can be simplified as (9)

Where

are the water head values at mesh points

.

And then the Dirichlet boundary condition is considered. The function value at discrete mesh points on boundary can be given firstly. Then combined with formula (8), all water head function values at the mesh points can be calculated. When the water head values at discrete points are known, according to equation (2) and (8), we can calculate seepage velocity at each mesh grid. The discrete forms of seepage velocity are shown as ,

,

.

(10)

At the boundary points, the following formulas can be obtained ,

,

(11)

If the boundary conditions and initial conditions are known in the given domain, the numerical approximation of unknown function values at discrete points can be calculated by the formulas (6), (8) (10) and (11).

4 ALGORITHM AND THE NUMERICAL EXAMPLE From the difference schemes of the above equations, we give the following algorithm and main codes. Input n0; Set h=1/n0; n=n0+1; A=eye(n*n*n);b=zeros(n*n*n,1);x0=b;x1=b; - 74 www.ivypub.org/MC

for k=1,…,n for i=1,…,n for j=1,…,n set m=(k-1)*n^2+(i-1)*n+j; if k>=2&&k<n&&i>=2&&i<n&&j>=2&&j<n; A(m,m)=-6;A(m,m-1)=1;A(m,m+1)=1; A(m,m-n)=1;A(m,m+n)=1;A(m,m-n^2)=1;A(m,m+n^2)=1; end if k==1 b(m)=0; elseif k==n b(m)=H((i-1)*h,(j-1)*h,1); end if i==1 b(m)=H(0,(j-1)*h,(k-1)*h); elseif i==n b(m)=H(1,(j-1)*h,(k-1)*h); end if j==1 b(m)=H((i-1)*h,0,(k-1)*h); elseif j==n b(m)=H((i-1)*h,1,(k-1)*h); end end end end x=A\b; clear i j k m For k=1,…, n for i=1,…, n for j=1,…, n set m=(k-1)*n^2+(i-1)*n+j; x0(m)=H((i-1)*h,(j-1)*h,(k-1)*h); x1(m)=H0((i-1)*h,(j-1)*h,(k-1)*h); end end end Output x0; x; Set B=zeros(n^3,3); B0=B; B0(:,1)=x0; B0(:,2)=x0; B0(:,3)=x1; B1=B0; for k=1,…,n for i=1,…,:n for j=1,…,:n m=(k-1)*n^2+(i-1)*n+j; if k>=2&&k<n&&i>=2&&i<n&&j>=2&&j<n B(m,1)=(x(m+n)-x(m-n))/(2*h); B(m,2)=(x(m+1)-x(m-1))/(2*h); B(m,3)=(x(m+n^2)-x(m-n^2))/(2*h);B1(m,1)=0;B1(m,2)=0; B1(m,3)=0; end end - 75 www.ivypub.org/MC

end end B1; B2=B1+B; B2-B0;B=B2; Set K=1; % K is the permeability Set C=-K*B;C0=-K*B0; % C,C0 are the velocity at discrete points clear i j k m Input K1 t0 Set N=0.3; % Porosity Set n2=n^3*((t0*n0)+1); % n2 is the number of the discrete points Set E=zeros(n2,1); T1=E; For t=1,…, (t0*n0)+1 for k=1,…, n for i=1,…, n for j=1,…, n set m=(t-1)*n*n*n+(k-1)*n*n+(i-1)*n+j; T1(m)=T0((i-1)*h,(j-1)*h,(k-1)*h,(t-1)*h/t0); n3=(k-1)*n^2+(i-1)*n+j; if t==1 then set E(m)=T0((i-1)*h,(j-1)*h,(k-1)*h,(t-1)*h/t0); end if t>1&&k>=2&&k<n&&i>=2&&i<n&&j>=2&&j<n Then set E(m)=(K1/(h*t0)+C(n3,3)/(2*t0*N))*E(m-n^2-n^3)+E(m+n^2-n^3)*(K1/(h*t0)-C(n3,3)/(2*t0*N))+E(m-n-n^3)*(K 1/(h*t0)+C(n3,1)/(2*t0*N))+E(m+n-n^3)*(K1/(h*t0)-C(n3,1)/(2*t0*N))+E(-n^3+m-1)*(K1/(h*t0)+C(n3,2)/(2*t0*N ))+E(m+1-n^3)*(K1/(h*t0)-C(n3,2)/(2*t0*N))+E(m-n^3)*(1-6*K1/(h*t0)); end if k==1 set E(m)=T0((i-1)*h,(j-1)*h,(k-1)*h,(t-1)*h/t0); elseif k==n then set E(m)=T0((i-1)*h,(j-1)*h,(k-1)*h,(t-1)*h/t0); end if i==1 set E(m)=T0((i-1)*h,(j-1)*h,(k-1)*h,(t-1)*h/t0); elseif i==n then set E(m)=T0((i-1)*h,(j-1)*h,(k-1)*h,(t-1)*h/t0); end if j==1 set E(m)=T0((i-1)*h,(j-1)*h,(k-1)*h,(t-1)*h/t0); elseif j==n then set E(m)=T0((i-1)*h,(j-1)*h,(k-1)*h,(t-1)*h/t0); end end end end end OUTPUT Then, based on the above algorithm, we solve the following 3D coupled model consisting of equations (12) and (13). - 76 www.ivypub.org/MC

(12)

(13)

Where

.

The analytical solution to equation (11) is

, and the computed analytical solution to

equation (13) is

.

In this example, the definite domain is given as the unit cube. The cube is equidistant subdivision, each side length is 12 equal parts, the time period is unit time and time step is 1/1800. The numerical results of the temperature T and water head value H are shown as follows. 8

the Approximate Value of Water Head

the Exact Value of Water Head

8 7 6 5 4 3 2 1 0

0

500

1000 1500 2000 Disctete Points

2500

7 6 5 4 3 2 1 0

0

500

1000 1500 2000 Disctete Points

2500

FIG. 1 COMPARISON OF THE EXACT AND APPROXIMATION VALUE OF THE WATER HEAD - 77 www.ivypub.org/MC

TABLE 1 COMPARISON OF THE EXACT AND APPROXIMATION VALUE OF

The exact value H 0.193854200372343 0.385019096078945 0.570842690696917 0.748747086718487 0.916264248262269 1.071070239693822 1.211017465169250 1.334164461848409 1.438802833461809 1.523480950586641 1.587024088842251

The approximate value H1 0.193883391857379 0.385076676916620 0.570927018974783 0.748855600447801 0.916393331735950 1.071215018405726 1.211171500284468 1.334319297162373 1.438947315539083 1.523600184442877 1.587097798339812

0 -1 -2 -3 -4 -5 -6 -7

-8 -8 -6 -4 -2 0 the Exact Value of Seepage Velocity in X(Y) Direction

the Approximate Value of Seepage Velocity in Z Direction

the Approximate Value of Seepage Velocity in X(Y) Direction

z 0.083333333333333 0.166666666666667 0.250000000000000 0.333333333333333 0.416666666666667 0.500000000000000 0.583333333333333 0.666666666666667 0.750000000000000 0.833333333333333 0.916666666666667

H(x,y,z) WITH x=1/3, y=1/6, h=1/12 Error H1-H 0.000029191485036 0.000057580837675 0.000084328277866 0.000108513729314 0.000129083473680 0.000144778711904 0.000154035115218 0.000154835313965 0.000144482077274 0.000119233856236 0.000073709497560

0

-2

-4

-6

-8

-10

-12 -15 -10 -5 0 the Exact Value of Seepage Velocity in Z Direction

FIG. 2 COMPARISON OF THE EXACT AND APPROXIMATION VALUE OF SEEPAGE VELOCITY TABLE 2 COMPARISON OF THE EXACT AND SOME APPROXIMATION VALUE OF T(x,y,z) WITH x=1/3, y=1/6, z=1/4,h=1/12, τ=1/1800, k2=0.01

Time t 0.0006 0.0172 0.0339 0.0506 0.0672 0.0839 0.1006 0.1172 0.1672 0.1839 0.2006 0.3506 0.3672 0.3839 0.4006 0.4172 0.4339 0.5506 0.6672 0.6839 0.7006 0.7172 0.8006 0.8172

The exact value T 0.3593 0.5311 0.7030 0.8749 1.0467 1.2186 1.3904 1.5623 2.0779 2.2497 2.4216 3.9684 4.1402 4.3121 4.4839 4.6558 4.8277 6.0307 7.2337 7.4056 7.5774 7.7493 8.6086 8.7804

The approximate value T1 0.3593 0.5308 0.7024 0.8740 1.0458 1.2181 1.3910 1.5645 2.0854 2.2590 2.4324 3.9936 4.1671 4.3407 4.5142 4.6877 4.8612 6.0762 7.2911 7.4646 7.6382 7.8120 8.6688 8.8194

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Error T1-T -0.0000 -0.0003 -0.0006 -0.0008 -0.0009 -0.0005 0.0006 0.0022 0.0075 0.0093 0.0108 0.0252 0.0269 0.0286 0.0303 0.0319 0.0336 0.0455 0.0574 0.0590 0.0607 0.0627 0.0602 0.0390

25

Approximate Value of the Water Temperature

Exact Value of the Water Temperature

25

20

15

10

5

0

0

0.5 1 1.5 Disctete Points

2

20

15

10

5

0

0

6

x 10

0.5 1 1.5 Disctete Points

2 6

x 10

FIG. 3 COMPARISON OF THE EXACT AND APPROXIMATION VALUE OF WATER TEMPERATURE

5 CONCLUSIONS This paper presented the difference scheme of the three-dimensional thermal-fluid coupled mathematic model and tested the given algorithm with the numerical example which has a large amount of calculation. The numerical simulation example was given in the special definite domain and the results of the compute test showed the efficiency of the algorithm. The numerical method was taken into consideration that can reduce the compute cost and the numerical simulation examples defined in the general domain.

ACKNOWLEDGMENTS The work was supported by the National Nature Science Foundation of China (Nos.11161002 and 41001320), Natural Science Foundation of Jiangxi province (No.20114BAB201016). Thanks for the useful advices of the editors and the reviewers.

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[2]

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[3]

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[4]

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[5]

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[10] Vineet K. Srivastava, Mukesh K. Awasthi, and Mohammad Tamsir. A Fully Implicit Finite-difference Solution to One Dimensional Coupled Nonlinear Burgers’ Equations[M]. International Journal of Mathematical Sciences, 7(4), 2013. [11] Shelly Mcgee, Padmanabhan Seshaiyer. Finite Difference Methods For Coupled Flow Interaction Transport Models[M]. Electronic Journal of Differential Equations, Conf. 17 (2009), 171–184. - 79 www.ivypub.org/MC

[12] Vineet Kumar Srivastava, Mohammad Tamsir, Utkarsh Bhardwaj. Crank-Nicolson Scheme for Numerical Solutions of Twodimensional Coupled Burgersâ€™ Equations. International Journal of Scientific & Engineering Research, 2(5), 2011.

AUTHORS 1

2

China, in 1987, male, Han nationality. He

graduated from the University of Chinese Academy of Sciences,

became a graduate student in school of

master instructor and associate professor of East China Institute

science,

Xianrui Chen was born in ShanXi,

East

Technology

in

computational interested

in

China 2011.

Tangwei Liu (1973-), male, the Han nationality, Ph.D.

Institute

of

of Technology. His research interests are in computational

His major

is

geodynamics and inverse problem computing.

mathematics. numerical

He

is

methods

of

Email: twliu@ecit.cn.

coupled mathematic models. Email: chenxiangrui12345@163.com.

- 80 www.ivypub.org/MC